Spin relaxation of electron and hole polarons in ambipolar conjugated polymers

The charge-transport properties of conjugated polymers have been studied extensively for opto-electronic device applications. Some polymer semiconductors not only support the ambipolar transport of electrons and holes, but do so with comparable carrier mobilities. This opens the possibility of gaining deeper insight into the charge-transport physics of these complex materials via comparison between electron and hole dynamics while keeping other factors, such as polymer microstructure, equal. Here, we use field-induced electron spin resonance spectroscopy to compare the spin relaxation behavior of electron and hole polarons in three ambipolar conjugated polymers. Our experiments show unique relaxation regimes as a function of temperature for electrons and holes, whereby at lower temperatures electrons relax slower than holes, but at higher temperatures, in the so-called spin-shuttling regime, the trend is reversed. On the basis of theoretical simulations, we attribute this to differences in the delocalization of electron and hole wavefunctions and show that spin relaxation in the spin shuttling regimes provides a sensitive probe of the intimate coupling between charge and structural dynamics.

teraction with the field manifests as the paramagnetic bulk becoming magnetized at equilibrium, a process which occurs over a relatively quick timescale.A continuous interaction is achieved by applying a oscillatory microwave field with amplitude B mw and frequency ω mw to the sample, which acts to repeatedly push the sample out of equilibrium each time it aligns.This repeated absorption of microwave energy is detected as the ESR signal.The characteristic timescale the sample takes to become unaligned with the quantization axis (set by the external field) is the longitudinal relaxation time, T 1 , while the characteristic timescale over which coherent spins no longer precess around the quantization axis with each other is the transverse relaxation time, T 2 .The power absorbed over one cycle of the microwave field can be written as follows: where χ 0 is the static magnetic susceptibility of the sample and is the half-width-at-half-height of the normalized Lorentzian function L .It is clear that absorption will be greatest when the resonance condition ω mw = ω L is fulfilled.
The previous expression describes the resonance signal if all spins can be treated equivalently.This may not the case, however, and a more general function is required.It is often assumed that a Gaussian spread of resonance positions can be used to describe a collection of spins in such a case: The overall lineshape then follows as a convolution of this Gaussian spread with the Lorentzian resonance signal, which is known as a Voigtian.This is normally taken as a more general function to fit, and the Gaussian spread is negligible in cases where all spins are indeed equivalent.
There are two experimental factors that must also be considered before fitting.The first is that the detection diode is proportional to the square root of the microwave power, which modifies the absorption equation so that it is only linear in the amplitude of the microwave field.The second is that a lock-in amplifier is used to increase sensitivity of the signal.The effect of a lock-in amplifier is to detect the derivative absorption signal with respect to the Zeeman field; this is mathematically equivalent to adding a sinusoidal modulation signal to the field.Care must be taken in sitting the amplitude of the modulation signal, as signals that are too large can create artifacts in the signal.To correct for this, a routine in MATLAB was used for the data described here.By exploiting the fact that the diode's response to the signal can be written as a Fourier transform of the signal, and that phase-sensitive detection has no variance with time, the routine models the recorded spectrum as a convolution between a modulation kernel and the absorption signal.The resultant signal we fit has the form where is defined by the nth-order Bessel function.This is the equation to which we fit the data in order to obtain the relaxation times T 1 and T 2 , as well as resonance positions and root-mean-square of the convolving Gaussian spread (if any).

Low-temperature spectra
In Fig. 1, we show ESR data for all three systems at 10 K and 100 K for comparison to the data presented in

Longitudinal and transverse relaxation times
The full temperature dependencies of T 1 and T 2 for electron and hole polarons in all three systems in shown in Fig. 2. As reported in the main text, all three systems show the inhomogeneous broadening, motional narrowing, and spin-shuttling regimes.Because T 1 monotonically decreases with temperature, only T 2 is shown in the main text.The third mode, θ k,k ′ , represents the torsion between successive monomer units k and k' along the polymer axis and modulates the nearest-neighbor electronic coupling J so that the non-local electron-phonon coupling is accounted for.θ eq. is the equilibrium value of the inter-monomer torsion angle in the chargeneutral polymer chain.I k is the moment of inertia of monomer k, ω k,3 is the angular velocity of the third mode, and K θ is the inter-monomer torsional stiffness constant.
We remark that the Hamiltonian in eq.7 is similar to the one used before to study intra-chain transport in P3HT polymers [4] and IDTBT copolymers.[3] The main difference here is related to the steps taken to parameterize elements of the Hamiltonian (which will be described in the following sections) and the presence of two modes modulating the local e-ph couplings.

Figure 2 Supplementary Figure 1 :
Figure2in the main text.It is important to note that differences in signal-to-noise ratios between materials may represent true material differences and/or differences in measurement conditions.The intensities of the resonance signals of both holes and electrons in DPPT-TT (Fig.1(a)) at 10 K are particularly low, thus resulting in the comparatively large, visible signal from background impurities mentioned in the previous supplementary note.Fitting these spectra was not an issue, as evidenced by the reasonable error bars shown for T 1 and T 2 at 10 K in Figure3(a) in the main text.

Supplementary Figure 2 :
Full temperature dependence of n-and p-type polaronic relaxation times.All three polymers studied show the three regimes of spin relaxation: inhomogeneous broadening, motional narrowing, and spin-shuttling.

Supplementary Figure 4 :
Charge distribution.(a): Localization of the neutral (dotted black line), radical cation P+ (dashed orange line), and radical anion P-(grey line) wavefunctions simulated for a chain of five repeat units.(b): Torsion angle of neutral, positive, and negative pentamers.